Question

In the formula 1/3 π r*2 •h, if r is doubled and h is divided by 3, what is the ratio of new value of the formula to the original value?

Answer 1

$\bf \textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=\stackrel{doubled}{2r}\\ h=\stackrel{\textit{divided by 3}}{\frac{h}{3}} \end{cases}\implies V=\cfrac{\pi (2r)^2\left( \frac{h}{3} \right)}{3} \\\\\\ V=\cfrac{\pi (2^2r^2)\left( \frac{h}{3} \right)}{3}\implies V=\cfrac{~~\frac{4\pi r^2 h}{3}~~}{3}\implies V=\cfrac{4\pi r^2 h}{3\cdot 3}\implies \stackrel{\frac{4}{3}\textit{ of the original}}{V=\cfrac{4}{3}\cdot \cfrac{\pi r^2 h}{3}}$